Hindawi Publishing Corporation Discrete Dynamics in Nature and Society Volume 2013, Article ID 320581, 10 pages http://dx.doi.org/10.1155/2013/320581 Research Article Analysis of a Dengue Disease Model with Nonlinear Incidence Shu-Min Guo,1 Xue-Zhi Li,2 and Mini Ghosh3 1 Department of Mathematics and Information Science, Shaoguan University, Shaoguan 512005, China Department of Mathematics, Xinyang Normal University, Xinyang 464000, China 3 School of Advanced Sciences, VIT University, Chennai Campus, Chennai 600048, India 2 Correspondence should be addressed to Xue-Zhi Li; xzli66@126.com Received 10 September 2012; Revised 3 January 2013; Accepted 3 January 2013 Academic Editor: Eric Campos Canton Copyright © 2013 Shu-Min Guo et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. A dengue disease epidemic model with nonlinear incidence is formulated and analyzed. The equilibria and threshold of the model are found. The stability of the system is analyzed through a geometric approach to stability. The proposed model also exhibits backward bifurcation under suitable conditions on parameters. Our results imply that a nonlinear incidence produces rich dynamics and they should be studied carefully in order to analyze the spread of disease more accurately. Finally, numerical simulations are presented to illustrate the analytical findings. 1. Introduction Dengue disease is transmitted in humans by mosquitoes, called vectors, who carry the disease without getting it themselves. Every year many people die from dengue disease. It has become a global health problem due to its rapid spread. In order to predict the development extent of such epidemics, it is useful to find the suitable mathematical models, which can provide insight into the dynamics of a disease and can help us to make right decisions on public health policies. However in the region of high infective prevalence such as Africa, some tropical countries, and so forth, there are lots of people infected with dengue, and so forth. and hence there are lots of infected mosquitoes. So in these endemic regions residents are frequently bitten by infected mosquitoes, thus making individuals exposed with multiple bites of mosquito which causes individuals to move to infected class with increased rate compared to people who are exposed with single effective bite of mosquito. So it is reasonable to consider nonlinear saturation incidence in place of bilinear incidence or standard incidence in the model as it also incorporates the effects of increased exposure to mosquito bites. In fact the incidence rate is very important in the transmission dynamics of the disease and the qualitative behaviour of the disease changes with the change in the incidence rate. In a recent paper by Xiao and Tang [1], interaction of the nonlinear incidence and the partial immunity for a simple SIV epidemic model is investigated and it is shown that due to the nonlinear incidence, vaccination may contribute to disease spread rather than to its elimination. A vector-host epidemic model with nonlinear incidence is also analyzed by Cai and Li [2]. Here they considered nonlinear saturating type incidence and they studied the effect of this type of incidence rate on the basic reproduction number and the equilibrium level of the infective population. Hu et al. [3] analyzed an SIR epidemic model with nonlinear incidence rate and treatment. Here it is shown that reducing the basic reproduction number below one is not enough to eliminate the disease as backward bifurcation occurs for this model. Also when the basic reproduction number is greater than unity, there are multiple endemic equilibria. The conditions for the stability of endemic equilibrium and the existence of limit cycle are discussed in this paper. A discrete-time epidemic model with nonlinear incidence rates is studied by Li et al. [4], where a very complex dynamical behaviors are demonstrated. It is shown that there are possibilities of the transcritical bifurcation, flip bifurcation, Hopf bifurcation and chaos. A vector-borne model with nonlinear incidence is analyzed by Ozair et al. [5]. Here they considered the conditions for the stability of endemic equilibrium. 2 Discrete Dynamics in Nature and Society In recent years, dengue has become a major public health problem and is endemic in several countries. It is affecting almost one-third of the world’s population. Dengue is an infection caused by a virus known as flavivirus and is spread by their vector Aedes aegypti. There are four different serotypes of flavivirus but infection with one serotype gives life-long immunity to that serotype but not to other three serotypes. So one can formulate dengue disease model as SIS or SIR model depending upon serotypes of flavivirus. In this paper we have formulated an SIR epidemic model with nonlinear incidence assuming the infection of dengue is caused by the flavivirus which is giving life-long immunity. In dengue-endemic area a person can be infected with more than one serotype of flavivirus in his/her lifetime. But in the current study we have ignored this fact as it will complicate the model and its analysis. The paper is organized as follows. In Section 2 the model and some preliminary properties are presented. In Section 3 equilibria and bifurcation are studied. Section 4 is devoted to the global stability analysis of the endemic equilibrium, when it is unique. Numerical simulations are given in Section 5, we end the paper with conclusions in Section 6. 2. Model Formulation The vector-host epidemic model with nonlinear incidence can be described by the following system of differential equations: ππ (π‘) = π1 − π 1 π (π‘) π (π‘) [1 + πΌπ (π‘)] − π1 π (π‘) , ππ‘ ππ1 (π‘) = π1 − π1 π1 (π‘) , ππ‘ (2) which is derived by adding the first three equations in (1). The total number of vectors π2 (π‘) can be determined by π2 (π‘) = π(π‘) + π(π‘) or from the differential equation ππΌ (π‘) = π 1 π (π‘) π (π‘) [1 + πΌπ (π‘)] − πΎπΌ (π‘) − π1 πΌ (π‘) , ππ‘ ππ (π‘) = πΎπΌ (π‘) − π1 π (π‘) , ππ‘ new infective individuals arise from double exposures at a rate πΌπ 1 π(π‘)π2 (π‘). The second component of the vector population is the number of pathogen-free (susceptible) vectors at time π‘, given by π(π‘). The total size of the vector population at time π‘, given by π2 (π‘) and is subdivided into these two vector-population classes, the susceptible vectors and infectious vectors. The vector population dies at a natural death rate π2 . In addition, the vector population is recruited at a birth rate π2 . Although there are evidences that the pathogen of several vectorborne diseases (e.g., West Nile fever, yellow fever and Lyme disease) can be transmitted from (female) parent to offspring in the vector population, we will assume that all newborn vectors are susceptible and vertical transmission is neglected. Susceptible vectors start carrying the pathogen after getting into contact (biting) an infective host at a rate π 2 so that the incidence of newly infected vectors is given by a mass-action term π 2 π(π‘)πΌ(π‘). In contrast to the host population, once the vectors become carriers of the microparasite, they carry it for life. We make some reasonable technical assumptions on the parameters of the model, namely that πΎ > 0, ππ > 0 and ππ > 0 for π = 1, 2. The above systems for the host population and the vector are also equipped with initial conditions as follows: π(0) = π0 , πΌ(0) = πΌ0 , π (0) = π 0 , π(0) = π0 and π(0) = π0 . The total population size π1 (π‘) can be determined by π1 (π‘) = π(π‘) + πΌ(π‘) + π (π‘) or from the differential equation ππ2 (π‘) = π2 − π2 π2 , ππ‘ (1) ππ (π‘) = π2 − π 2 π (π‘) πΌ (π‘) − π2 π (π‘) , ππ‘ ππ (π‘) = π 2 π (π‘) πΌ (π‘) − π2 π (π‘) . ππ‘ The total host population size at time π‘, given by π1 (π‘), is partitioned into subclasses of individuals who are susceptible, infectious and recovered, with sizes denoted by π(π‘), πΌ(π‘) and π (π‘), respectively. Furthermore, the host population dies at a natural death rate π1 . In addition, the host population is recruited at a rate π1 . We assume that vertical transmission in the host does not occur so that all newly recruited individuals are susceptible. The per capita recovery rate of the hosts is given by πΎ. The recovered individuals are assumed to acquire permanent immunity and there is no transfer from the π class back to the π class. The nonlinear incidence rate is π 1 π(π‘)π(π‘)[1 + πΌπ(π‘)], where π 1 and πΌ are positive constants, π(π‘) is the number of vectors at time π‘ who carry the pathogen. It corresponds to an increased rate of infection due to two exposures over a short time period. The single contacts lead to infection at the rate π 1 π(π‘)π(π‘), whereas the (3) which is derived by adding the last two equations in (1). It is easily seen that both for the host population and for the vector population the corresponding total population sizes are asymptotically constant: lim π1 (π‘) = π‘→∞ π1 , π1 lim π2 (π‘) = π‘→∞ π2 . π2 (4) So we can assume without loss of generality that π1 (π‘) = π1 /π1 , π2 (π‘) = π2 /π2 for all π‘ ≥ 0. Then the dynamics of system (1) is qualitatively equivalent to the dynamics of system given by ππ (π‘) = π1 − π 1 π (π‘) π (π‘) [1 + πΌπ (π‘)] − π1 π (π‘) , ππ‘ ππΌ (π‘) = π 1 π (π‘) π (π‘) [1 + πΌπ (π‘)] − πΎπΌ (π‘) − π1 πΌ (π‘) , ππ‘ (5) π ππ (π‘) = π 2 [ 2 − π (π‘)] πΌ (π‘) − π2 π (π‘) . ππ‘ π2 The values of π and π can be determined correspondingly from π = π1 /π1 − π − πΌ, and π = π2 /π2 − π respectively. Discrete Dynamics in Nature and Society 3 For biological reasons we need the solutions nonnegative. Mathematical properties of the solutions lead us to study the system (5) in the closed set Γ = {(π, πΌ, π) ∈ R+3 | 0 ≤ π + πΌ ≤ π1 π , 0 ≤ π ≤ 2 , π ≥ 0, πΌ ≥ 0} , π1 π2 (6) where R+3 denotes the non-negative cone of R3 including its lower dimensional faces. It can be verified that Γ is positively invariant with respect to (5). We denote by πΓ and Γ0 the boundary and the interior of Γ in R3 , respectively. Furthermore, we observe that there is a bifurcation point when π3 > 0, π2 < 0 and π22 − 4π1 π3 = 0. In fact the quantity π22 − 4π1 π3 can be expressed in terms of new threshold π 0∗ , where π 0∗ = π0 = π0 , π0 2π21 π22 π12 π2 πΌ 2π2 π π π πΌ + 1 2 1 2 + π0 , π1 π1 π2 (πΎ + π1 ) π 1 π 2 π1 π22 + 2π 1 π23 (πΎ + π1 ) π 1 π24 (πΎ + π1 ) + π2 π 2 π1 π2 3. Equilibria and Bifurcation Analysis + π 1 π 2 π1 π2 πΌ2 + π0 , The disease-free equilibrium of system (5) is πΈ0 = (π1 / π1 , 0, 0). The endemic equilibria πΈ1 = (π∗ , πΌ∗ , π∗ ) of system (5) can be deduced by the system, π0 = 4π 1 π 2 π1 πΌπ2 + 4π 1 πΌπ22 (πΎ + π1 ) . π1 − π 1 π∗ π∗ [1 + πΌπ∗ ] − π1 π∗ = 0, π 1 π∗ π∗ [1 + πΌπ∗ ] − πΎπΌ∗ − π1 πΌ∗ = 0, (7) π π 2 [ 2 − π∗ ] πΌ∗ − π2 π∗ = 0, π2 which gives, π∗ = π1 , π 1 π∗ (1 + πΌπ∗ ) + π1 πΌ∗ = π2 π∗ , π 2 (π2 /π2 − π∗ ) π1 π ∗ + π2 π + π3 = 0, By a little algebraic calculation, we have π22 − 4π1 π3 > 0 whenever π 0 > π 0∗ . Taking into account the above consideration, we have the following theorem. Theorem 1. System (7) admits a unique endemic equilibrium πΈ when π 0 ≥ 1; there are no endemic equilibria for π 0 < π 0∗ ; there are two endemic equilibria, πΈ1 and πΈ2 , for π < π 0 < 1 and π = max{π 0∗ , π 1∗ }. Next we analyze the stability of the disease-free equilibrium πΈ0 = (π1 /π1 , 0, 0). The Jacobian matrix corresponding to (7) is (8) and π∗ is the positive solution of the following equation, ∗2 π½ (π, πΌ, π) −π 1 π (1 + πΌπ)−π1 0 −π 1 π (1+2πΌπ) π 1 π (1+πΌπ) −πΎ−π1 π 1 π (1+2πΌπ) (9) =( where 0 π1 = π 1 π 2 π1 πΌ + π2 π 1 πΌ (πΎ + π1 ) , π2 = π 1 π 2 π1 + π2 π 1 (πΎ + π1 ) − π2 ( π 1 π 2 π1 π2 πΌ π2 π2 −π) π2 π 1∗ = (13) π 1 π 2 π1 π2 = π1 π2 (πΎ + π1 ) (1 − π 0 ) , π2 π 1 π 2 π1 + π2 π 1 (πΎ + π1 ) , πΌπ1 π2 (πΎ + π1 ) π 0 = π 1 π 2 π1 π2 . π1 π22 (πΎ + π1 ) π2 < 0 ⇐⇒ π 1∗ < π 0 ; π3 > 0 ⇐⇒ π 0 < 1. First, we observe that −π1 π½( (10) π1 ( , 0, 0) = ( 0 π1 ( 0 0 −πΎ − π1 π2 π2 π2 π1 π1 π1 ) π1 ), π1 −π 1 −π2 (14) ) so that the eigenvalues π are given by the roots of the following qubic equation Thus, we observe that π1 > 0; ). −π 2 πΌ−π2 = πΌπ1 π2 (πΎ + π1 ) (π 1∗ − π 0 ) , π3 = π1 π2 (πΎ + π1 ) − (12) 2 π0 = (11) By the Descartes’ rule of signs, it follows that there is a unique endemic equilibrium whenever π3 < 0 and there are two endemic equilibria whenever π3 > 0, π2 < 0 and π22 − 4π1 π3 > 0. (π1 + π) [π2 + (πΎ + π1 + π2 ) π + π2 (πΎ + π1 ) (1 − π 0 )] = 0. (15) Hence if π 0 < 1, then the eigenvalues are all negatives and πΈ0 is locally asymptotically stable. If π 0 > 1, then two 4 Discrete Dynamics in Nature and Society eigenvalues are negative and one is positive, so that πΈ0 is unstable. Since there are no explicit expressions for equilibria πΈ1 and πΈ2 and the existence conditions for the two equilibria are very complex, it is very difficult to prove their stability by using the existing analytic methods. Theorem 1 establishes that π 0 = 1 is a bifurcation value. In fact, across π 0 = 1 the disease free equilibrium changes its stability properties. Now, we investigate what kind of bifurcation occurs at π 0 = 1. In order to do that, we will make use of the result summarized below, which has been obtained in [6] and is based on the use of the center manifold theory [7]. Let us consider a general system of ODEs with a parameter π: π π₯Μ = π (π₯, π) ; π π : π × π σ³¨→ π , 2 It clearly appears that, at π = 0 a transcritical bifurcation takes place: more precisely, when π < 0 and π > 0, such a bifurcation is forward; when π > 0 and π > 0 the bifurcation at π = 0 is backward. We will apply Theorem 2 to show that system (7) may exhibit a backward bifurcation when π 0 = 1. We consider the disease-free equilibrium πΈ0 = (π1 /π1 , 0, 0) and observe that the condition π 0 = 1 can be seen, in terms of the parameter π 1 , as π 1 = π∗ = π1 π22 (πΎ + π1 )/π 2 π1 π2 . The eigenvalues of the matrix −π1 ( π½ (πΈ0 , π∗ ) = ( 0 ( π π ∈ πΆ (π × π ) . (16) Without loss of generality, we assume that π₯ = 0 is an equilibrium for (16). Theorem 2 (see (Castillo-Chavez and Song [6])). Assume: (A1) π΄ = π·π₯ π(0, 0) is the linearization matrix of system (16) around the equilibrium π₯ = 0 with π evaluated at 0. Zero is a simple eigenvalue of π΄ and all other eigenvalues of π΄ have negative real parts. π1 = −π1 , π = ∑ ππ ππ ππ π,π,π π ππ (0, 0) , ππ₯π ππ₯π π π = ∑ ππ ππ π,π,π (iii) π > 0, π < 0. When π < 0, with |π| βͺ 1, π₯ = 0 is unstable and there exists a locally asymptotically stable negative equilibrium; when 0 < |π| βͺ 1, π₯ = 0 is stable and a positive unstable equilibrium appears; (iv) π < 0, π > 0. When π changes from negative to positive, π₯ = 0 changes its stability from stable to unstable. Correspondingly, a negative unstable equilibrium becomes positive and locally asymptotically stable. ) π3 = 0. (19) π22 (πΎ + π1 ) π3 = 0, π 2 π2 (20) π 2 π2 π − π2 π3 = 0, π2 2 (17) (ii) π < 0, π < 0. When π < 0, with |π| βͺ 1, π₯ = 0 is unstable; when 0 < |π| βͺ 1, π₯ = 0 is locally asymptotically stable and there exists a positive unstable equilibrium; −π2 π22 (πΎ + π1 ) π3 = 0, π 2 π2 − (πΎ + π1 ) π2 + π ππ (0, 0) . ππ₯π ππ (i) π > 0, π > 0. When π < 0, with |π| βͺ 1, π₯ = 0 is locally asymptotically stable and there exists a positive unstable equilibrium; when 0 < |π| βͺ 1, π₯ = 0 is unstable and there exists a negative and locally asymptotically stable equilibrium; π2 π2 π2 = −πΎ − π1 − π2 , −π1 π1 − 2 Then the local dynamics of system (16) around π₯ = 0 are totally determined by π and π π2 (18) Thus π3 = 0 is a simple zero eigenvalue of the matrix π½(πΈ0 , π∗ ) and the other eigenvalues are real and negative. Hence, when π 1 = π∗ (or equivalently when π 0 = 1), the disease-free equilibrium πΈ0 is a nonhyperbolic equilibrium: the assumption (A1) of Theorem 2 is then verified. Now we denote by π = (π1 , π2 , π3 )π , a right eigenvector associated with the zero eigenvalue π3 = 0. It follows: Let ππ denotes the πth component of π and, 2 −πΎ − π1 0 π1 π1 π π∗ 1 ) ), π1 −π∗ are given by (A2) Matrix π΄ has a right eigenvector π and a left eigenvector π corresponding to the zero eigenvalue. π 0 so that π π = (− πΎ + π1 π π , 1, 22 2 ) . π1 π2 (21) Furthermore, the left eigenvector π = (π1 , π2 , π3 ) satisfying π ⋅ π = 1 is given by: −π1 π1 = 0, − (πΎ + π1 ) π2 + π 2 π2 π = 0, π2 3 π22 (πΎ + π1 ) (π2 − π1 ) − π2 π3 = 0, π 2 π2 π3 ( (22) π 2 π2 π π + 22 2 ) = 1. π2 π2 (πΎ + π1 ) Then, the left eigenvector π turns out to be: π = (0, π22 (πΎ + π1 ) π2 , ). πΎ + π1 + π2 π 2 π2 (πΎ + π1 + π2 ) (23) Discrete Dynamics in Nature and Society 5 We can thus compute the coefficient π and π defined in Theorem 2, that is, 3 π2 ππ π = ∑ ππ ππ ππ (πΈ0 , π∗ ) , ππ₯ ππ₯ π π π,π,π=1 3 π2 ππ π = ∑ ππ ππ (πΈ0 , π∗ ) . ππ₯ ππ π 1 π,π=1 (24) Taking into account of system (5) and considering in π and π only the nonzero derivatives for the terms (π2 ππ / ππ₯π ππ₯π )(πΈ0 , π∗ ) and (π2 ππ /ππ₯π ππ 1 )(πΈ0 , π∗ ), it follows that: π = 2π1 π1 π3 + 2π3 π2 π3 lim inf π (π‘) > π, π‘ → +∞ π2 π3 (πΈ , π∗ ) , ππππΌ 0 lim inf πΌ (π‘) > π, π‘ → +∞ (28) lim inf π (π‘) > π. π‘ → +∞ (25) π2 π1 π2 π1 (πΈ0 , π∗ ) + π1 π3 (πΈ , π∗ ) ππππ 1 ππππ 1 0 + π2 π1 In Section 3 it has been shown that π 0 ≥ 1 implies the existence and uniqueness of the endemic equilibrium πΈ. The stability analysis of πΈ will be here performed through the geometric approach to global stability due to Li and Muldowney [8]. The method has been summarized in the Appendix. System (5), under the assumption π 0 > 1, satisfies conditions (H1) and (H2) in Appendix. In fact, when π 0 > 1, then πΈ0 is unstable. The instability of πΈ0 , together with πΈ0 ∈ πΓ, implies the uniform persistence, that is, there exists a constant π > 0 such that: π2 π1 π2 π (πΈ0 , π∗ ) + π1 π32 21 (πΈ0 , π∗ ) ππππ ππ π2 π2 π2 π + 2π2 π1 π3 (πΈ0 , π∗ ) + π2 π32 22 (πΈ0 , π∗ ) ππππ ππ π = π1 π1 4. Global Stability Analysis π2 π2 π2 π2 (πΈ0 , π∗ ) + π2 π3 (πΈ , π∗ ) . ππππ 1 ππππ 1 0 The uniform persistence is equivalent to the existence of a compact set in the interior of Γ which is absorbing for (5). Thus, (H1) is verified. Moreover, as previously shown, πΈ is the only equilibrium in the interior of Γ, so that (H2) is verified, too. Now it remains to find conditions for which the Bendixson criterion given by Theorem A.4 is verified. Taking into account of the Jacobian matrix (13), we obtain the second additive compound matrix π½[2] (π, πΌ, π), [2] π½11 In view of (21) and (23), we get π½[2] 2π 2 π= [π π π π2 πΌ 3 π1 π2 (πΎ + π1 + π2 ) 1 2 1 2 π2 = (π 2 ( − π) π2 0 π 1 π (1 + 2πΌπ) π 1 π (1 + 2πΌπ) [2] π½22 0 π 1 π (1 + πΌπ) [2] π½33 (29) − π22 (πΎ + π1 ) (π 1 π2 + π1 π2 )] , π= π 2 π1 π2 . π1 π2 (πΎ + π1 + π2 ) where πΏ= π22 (πΎ + π1 ) (π 1 π2 + π1 π2 ) [2] π½11 = −π 1 π (1 + πΌπ) − πΎ − 2π1 , (26) Observe that the coefficient π is always positive so that, according to Theorem 2, it is the sign of the coefficient π which decides the local dynamics around the disease-free equilibrium for π 1 = π∗ . Let us introduce π 1 π 2 π1 π22 πΌ ), . [2] = −π 1 π (1 + πΌπ) − π1 − π 2 πΌ − π2 , π½22 [2] = −πΎ − π1 − π 2 πΌ − π2 . π½33 We consider the following function πΌ πΌ πΌ π = π (π, πΌ, π) = diag { , , } . π π π (27) The coefficient π is positive if and only if πΏ > 1. In this case, the direction of the bifurcation of system (5) at π 0 = 1 is backward. So we summarize the above results in the following theorem. Theorem 3. If πΏ > 1, system (5) exhibits a backward bifurcation when π 0 = 1. If πΏ < 1, system (5) exhibits a forward bifurcation when π 0 = 1. The local dynamics in the neighborhood of the bifurcation value π 0 = 1 is described in the former case by the (i) of Theorem 2 and in the later case by the (iv) of the same theorem. (30) (31) So that ππ = diag { ππ π −1 πΌ σΈ π − πσΈ πΌ πΌ σΈ π − πσΈ πΌ πΌ σΈ π − πσΈ πΌ , , }, π2 π2 π2 πΌ σΈ πσΈ πΌσΈ πσΈ πΌσΈ πσΈ = diag { − , − , − } . πΌ π πΌ π πΌ π (32) Therefore π΅ π΅ π΅ = ππ π−1 + ππ½[2] π−1 = ( 11 12 ) , π΅21 π΅22 (33) 6 Discrete Dynamics in Nature and Society where π΅11 = Substituting (40) into (38) and (40) into (39), respectively, we have πΌσΈ π π πΌ π1 = − 2 2 + π 2 πΌ + π2 − π 1 π (1 + πΌπ) πΌ π2 π πΌσΈ πσΈ − − π 1 π (1 + πΌπ) − 2π1 − πΎ, πΌ π π΅12 = [π 1 π (1 + 2πΌπ) , π 1 π (1 + 2πΌπ)] , π΅21 = [π 2 ( − 2π1 − πΎ + π 1 π (1 + 2πΌπ) π π2 − π) , 0] , π2 ≤ π΅22 πΌσΈ πσΈ 0 [ − −π 1 π (1+πΌπ)−π1 −π 2 πΌ−π2 ] πΌ π ]. =[ [ ] πΌσΈ πσΈ π 1 π (1 + πΌπ) − −πΎ−π1 −π 2 πΌ−π2 [ ] πΌ π π2 = ≤ (34) (35) σ΅¨ σ΅¨ σ΅¨ σ΅¨ π (π΅) ≤ sup {π1 , π2 } = sup {π1 (π΅11 )+ σ΅¨σ΅¨σ΅¨π΅12 σ΅¨σ΅¨σ΅¨ , π1 (π΅22 )+ σ΅¨σ΅¨σ΅¨π΅21 σ΅¨σ΅¨σ΅¨} , (36) where |π΅12 |, |π΅21 | are matrix norms with respect to the π1 vector norm, and π1 denotes the Lozinskii Μ measure with respect to the π1 norm. More specifically, π + π 1 π (1 + 2πΌπ) , π 2 ( 2 − π)} . π2 π1 π π + π2 + π 1 1 (1 + 2πΌ 2 ) < π 1 π (1 + πΌπ) + π1 + πΎ, π1 π1 π2 (44) π (π΅) ≤ πΌσΈ + Π, πΌ (45) where Π = max {π 2 π1 π π + π2 + π 1 1 (1 + 2πΌ 2 ) − π 1 π (1 + πΌπ) π1 π1 π2 − π1 − πΎ, π 2 ( πΌσΈ πσΈ π1 = − − π 1 π (1 + πΌπ) − 2π1 − πΎ + π 1 π (1 + 2πΌπ) , πΌ π (38) (39) Observe that system (5) provides the following equalities πσΈ π 2 π2 πΌ = − π 2 πΌ − π2 . π π2 π (43) In view of π ≤ π, πΌ ≤ π1 /π1 and π ≤ π ≤ π2 /π2 , we can deduce that if π π 2 ( 2 − π) < π1 , π2 (37) πΌσΈ πσΈ − − π1 − π 2 πΌ − π2 . πΌ π π πΌσΈ πσΈ − − π1 − π 2 πΌ − π2 + π 2 ( 2 − π) . πΌ π π2 (42) Λ = max {π 2 πΌ + π2 − π 1 π (1 + πΌπ) − π1 − πΎ Therefore, π2 = πΌσΈ − π1 + Λ, πΌ then 2 π1 (π΅22 ) = (41) where π2 πΌσΈ πσΈ − − π 1 π (1 + πΌπ) − πΎ − 2π1 , πΌ π σ΅¨σ΅¨ σ΅¨σ΅¨ σ΅¨σ΅¨π΅12 σ΅¨σ΅¨ = π 1 π (1 + 2πΌπ) , π σ΅¨σ΅¨ σ΅¨σ΅¨ σ΅¨σ΅¨π΅21 σ΅¨σ΅¨ = π 2 ( 2 − π) , π π πΌσΈ − π1 + π 2 ( 2 − π) . πΌ π2 π (π΅) ≤ where (π’1 , π’2 , π’3 ) denotes the vector in π 3 and denote by π the Lozinskii Μ measure with respect to this norm. It follows π1 (π΅11 ) = π πΌσΈ π 2 π2 πΌ − − π1 + π 2 ( 2 − π) πΌ π2 π π2 Thus, (36) implies Consider now the norm in π 3 as σ΅¨ σ΅¨ σ΅¨ σ΅¨ σ΅¨ σ΅¨ σ΅¨ σ΅¨σ΅¨ σ΅¨σ΅¨π’1 , π’2 , π’3 σ΅¨σ΅¨σ΅¨ = max {σ΅¨σ΅¨σ΅¨π’1 σ΅¨σ΅¨σ΅¨ , σ΅¨σ΅¨σ΅¨π’2 σ΅¨σ΅¨σ΅¨ + σ΅¨σ΅¨σ΅¨π’3 σ΅¨σ΅¨σ΅¨} , πΌσΈ + π 2 πΌ + π2 − π 1 π (1+πΌπ) − 2π1 − πΎ + π 1 π (1+2πΌπ) , πΌ (40) π2 − π) − π1 } < 0. π2 (46) Hence 1 πΌ (π‘) 1 π‘ + Π, ∫ π (π΅) ππ ≤ log π‘ 0 π‘ πΌ (0) (47) and the Bendixson criterion given by Theorem A.4 is thus verified. The discussions above may be summarized as follows. Theorem 4. If π 0 ≥ 1, then system (5) admits a unique endemic equilibrium πΈ. It is globally asymptotically stable with respect to solutions of (5) initiating in the interior of Γ, provided that inequality (44) holds true. Discrete Dynamics in Nature and Society 7 3000 2000 2500 1500 2000 πΌ πΌ 1500 1000 1000 500 500 0 0 500 1000 π 1500 0 2000 Figure 1: Phase plot of πΌ verses π showing bistability when π < π 0 < 1 and πΏ > 1 for the parameter values π1 = 240, π 1 = 0.00012, πΌ = 0.08, π1 = 0.16, πΎ = 0.015, π 2 = 0.0001, π2 = 550, π2 = 0.5. 0 500 1000 1500 π 2000 2500 3000 Figure 3: Phase plot of πΌ verses π showing stability of unique feasible disease free equilibrium point when π 0 < π 0∗ < 1 for the parameter values π1 = 80, π 1 = 0.00001, πΌ = 0.04, π1 = 0.06, πΎ = 0.015, π 2 = 0.0002, π2 = 200, π2 = 0.5. 3000 1000 2500 800 2000 πΌ πΌ 1500 400 1000 500 0 600 200 0 500 1000 1500 2000 π Figure 2: Phase plot of πΌ verses π showing stability of unique endemic equilibrium point when π 0 > 1 and πΏ > 1 for the parameter values π1 = 80, π 1 = 0.0004, πΌ = 0.04, π1 = 0.06, πΎ = 0.015, π 2 = 0.0002, π2 = 200, π2 = 0.5. 0 0 500 1000 1500 2000 π Figure 4: Phase plot of πΌ verses π showing stability of unique feasible endemic equilibrium point and instablity of disease free equilibrium point when πΏ < 1, π 0 > 1 for the parameter values π1 = 10, π 1 = 0.0005, πΌ = 0.01, π1 = 0.01, πΎ = 0.01, π 2 = 0.0003, π2 = 60, π2 = 0.5. 5. Simulation The system (5) is simulated for various sets of parameters using the package XPP [9]. In Figures 1–4, (π, πΌ) phase planes are drawn which confer the existence and the stability of different equilibria of the system (5). Here Figure 1 corresponds to the situation stated in Theorem 1, where π < π 0 < 1 and we get two endemic equilibria πΈ1 and πΈ2 and one disease free equilibria πΈ0 (1500, 0, 0). And for πΏ > 1, it is found that disease free equilibrium πΈ0 (1500, 0, 0) and the endemic equilibrium point with largest infective population πΈ2 (557.678, 861.551, 161.682) are stable and the equilibrium point πΈ1 (1170.057, 301.662, 62.589) is unstable. Figure 2 corresponds to the situation when π 0 > 1 and also πΏ > 1 and in this case we get unique stable endemic equilibrium point and unstable disease free equilibrium point πΈ0 . Figure 3 shows the stability of unique feasible disease free equilibrium πΈ0 (1333.3, 0, 0) for π 0 < π 0∗ < 1. For πΏ < 1, and π 0 ≥ 1 we get two feasible equilibria: the unstable disease free equilibria and a unique endemic equilibrium which is shown in Figure 4. This fact is more clear from the bifurcation Figure 5, which is obtained by considering π 1 as the critical parameter. The horizontal axis is labelled with the appropriate value of the reproduction number π 0 corresponding to this bifurcation parameter π 1 . For 0 < π 0 < 1 we get stable infection free equilibrium point πΈ0 and at π 0 = 1 we get forward bifurcation, that is, for π 0 > 1 we get unstable disease free equilibrium and unique stable endemic equilibrium point. Figure 6 is showing the backward bifurcation for the parameter values satisfying πΏ > 1. This diagram too is 8 Discrete Dynamics in Nature and Society Infective population (πΌ) 1400 1200 1000 800 600 400 200 0 0 0.5 1 1.5 Reproduction number (π 0 ) 2 Stable Figure 5: Forward bifurcation diagram for πΏ < 1 for the parameter values π1 = 10, π 1 = 0.0005, πΌ = 0.01, π1 = 0.01, πΎ = 0.01, π 2 = 0.0003, π2 = 60, π2 = 0.5. Appendix Let π₯ σ³¨→ π(π₯) ∈ π π be a πΆ1 function for π₯ in an open set π· ⊂ π π . Consider the differential equation 1400 Infective population (πΌ) the disease dies out. We find the condition under which the system exhibits backward bifurcation at π 0 = 1. The stability of two endemic equilibria in the case of backward bifurcation for π 0 < 1 is not discussed analytically but we have shown these by numerical simulations. It is found that the endemic equilibrium with the largest infective population is stable, while the endemic equilibrium with less infective population is unstable. In the case of a unique endemic equilibrium, the global stability analysis has been performed. A generalization of the PoincareΜ-Bendixson criterion has been used. The sufficient conditions that we found are not completely satisfactory, as they involve the constant of uniform persistence π. How to implement the geometric approach to give better stability conditions for epidemic models with convex incidence rate appears to be an open and stimulating challenge. 1200 π₯σΈ = π (π₯) . 1000 Denote by π₯(π‘, π₯0 ) the solution of (A.1) such that π₯(0, π₯0 ) = π₯0 . We take the following two assumptions. 800 600 (H1) There exits a compact absorbing set πΎ ⊂ π·. 400 (H2) Equation (A.1) has a unique equilibrium π₯ in π·. 200 0 0 0.2 0.4 0.6 0.8 1 1.2 Reproduction number (π 0 ) 1.4 1.6 1.8 Stable Unstable Figure 6: Backward bifurcation diagram for πΏ > 1 for the parameter values π1 = 240, π 1 = 0.00012, πΌ = 0.08, π1 = 0.16, πΎ = 0.015, π 2 = 0.0001, π2 = 550, π2 = 0.5. obtained by considering π 1 as the critical parameter and horizontal axis is labelled with the appropriate value of the reproduction number π 0 corresponding to this bifurcation parameter π 1 . It is observed that when the reproduction number π 0 is between 0 to 0.190526, the infection free equilibrium alone is stable, for 0.1905261 < π 0 < 1 we have bistability where either the infection free equilibrium is stable or the equilibrium πΈ2 is stable. Again by increasing π 1 we get π 0 > 1 and we have unique stable endemic equilibrium point and unstable disease free equilibrium πΈ0 . The equilibrium πΈ1 when it exists is always saddle. As backward bifurcation is quite significant in this figure, so it seems that it can arise for biologically reasonable parameter values too. 6. Conclusions In this paper, a dengue disease epidemic model with nonlinear incidence has been analyzed. If π 0 < 1, the diseasefree equilibrium πΈ0 is locally asymptotically stable, that is, (A.1) The equilibrium π₯ is said to be globally stable in π· if it is locally stable and all trajectories in π· converge to π₯. The following global-stability problem is formulated in [8]. Theorem A.1. Under the assumptions (H1) and (H2), the global stability of π₯ with respect to π· is implied by its local stability. Assumptions (H1) and (H2) are satisfied if π₯ is global stability in π·. For π ≥ 2, a Bendixson criterion is a condition satisfied by π which precludes the existence of nonconstant periodic solution of (A.1). A Bendixson criterion is said to be robust under πΆ1 local perturbations of π at π₯1 ∈ π· if, for sufficiently small π ≥ 0 and neighborhood π of π₯1 , it is also satisfied by π ∈ πΆ1 (π· → π π ) such that the support supp(π − π) ⊂ π and |π − π|πΆ1 < π, where σ΅¨ σ΅¨σ΅¨ ππ σ΅¨ σ΅¨σ΅¨ ππ σ΅¨ σ΅¨ σ΅¨σ΅¨ σ΅¨ (π₯)σ΅¨σ΅¨σ΅¨ : π₯ ∈ π·} . σ΅¨σ΅¨π − πσ΅¨σ΅¨σ΅¨πΆ1 = sup {σ΅¨σ΅¨σ΅¨π (π₯) − π (π₯)σ΅¨σ΅¨σ΅¨ + σ΅¨σ΅¨σ΅¨ (π₯)− σ΅¨σ΅¨ ππ₯ σ΅¨σ΅¨ ππ₯ (A.2) Such π will be called local π-perturbations of π at π₯1 . It is easy to see that the classic Bendixson’s condition div π(π₯) < 0 for π = 2 is robust under πΆ1 local perturbations of at each π₯1 ∈ π 2 . Bendixson’s condition for higher dimensional system that are πΆ1 robust are discussed in [8, 10, 11]. A point π₯0 ∈ π· is wandering for (A.1) if there exists a neighborhood π of π₯0 and π > 0 such that π ∩ π₯(π‘, π) is empty for all π‘ > π. Thus, for example, all equilibria and limit points are nonwandering. The following is a version of the local πΆ1 Closing lemma of Pugh (see [12, 13]). Discrete Dynamics in Nature and Society 9 Lemma A.2. Let π ∈ πΆ1 (π· → π π ). Assume that all positive semitrajectories of (A.1) are bounded. Suppose that π₯0 is a nonwandering point of (A.1) and that π(π₯0 ) =ΜΈ 0. Then, for each neighborhood π of π₯0 and π > 0, there exists a πΆ1 local-π perturbation π of π at π₯0 such that (a) supp(π − π) ⊂ π and Let π₯ σ³¨→ π(π₯) ( π2 ) × ( π2 ) matrix-valued function that is πΆ for π₯ ∈ π·. Assume that π−1 (π₯) exists and is continuous for π₯ ∈ πΎ, the compact absorbing set. A quantity π2 is defined as 1 1 π‘ π2 = lim sup sup ∫ π (π΅ (π₯ (π , π₯0 ))) ππ , π‘ → ∞ π₯0 ∈πΎ π‘ 0 where (b) the perturbed system π₯σΈ = π(π₯) has a nonconstant periodic solution whose trajectory passes through π₯0 . The following general global-stability principle is established in [8]. Theorem A.3. Suppose that assumptions (H1) and (H2) hold. Assume that (A.1) satisfies a Bendixson criterion that is robust under πΆ1 local perturbations of π at all nonequilibrium nonwandering points for (A.1). Then π₯ is globally stable in π· provided it is stable. The main idea of the proof in [8] for Theorem A.3 is as follows: suppose that system (A.1) satisfies a Bendixson criterion. Then it does not have any nonconstant periodic solutions. Moreover, the robustness of the Bendixson criterion implies that all nearby differential equations have nonconstant periodic solutions. Thus by Lemma A.2, all of nonwandering points of (A.1) in π· must be equilibria. In particular, each omega limit point in π· must be an equilibrium. Therefore π(π₯0 ) = {π₯} for all π₯0 ∈ π· since π₯ is the only equilibrium in π·. A method of deriving a Bendixson criterion in π π is developed in [14]. The idea is to show that second compound equation π§σΈ (π‘) = ππ[2] (π₯ (π‘, π₯0 )) π§ (π‘) , ππ₯ (A.3) with respect to a solution π₯(π‘, π₯0 ) ⊂ π· to (A.1), is uniformly asymptotically stable, and the exponential decay rate of all solutions to (A.3) is uniform for π₯0 in each compact subset of π·. Here ππ[2] /ππ₯ is second additive compound matrix of the Jacobian matrix ππ/ππ₯. Generally speaking, for an π × π matrix π½ = (π½ππ ), π½[2] is a ( π2 ) × ( π2 ) matrix and in the special case π = 3, one has π½[2] = ( π½11 + π½22 π23 −π½13 π½32 π½11 + π½33 π½12 ) . −π½31 π½21 π½22 + π½33 (A.5) (A.4) ππ[2] /ππ₯ is a ( π2 ) × ( π2 ) matrix, and thus (A.3) is a linear system of dimension ( π2 ). If π· is simply connected, the above mentioned stability property of (A.3) implies the exponential decay of the surface area of any compact 2π surface in π·, which in turn precludes the existence of any invariant simply closed rectifiable curve in π·, including periodic orbits. The required uniform asymptotic stability of system (A.3) can be proved by constructing a suitable Lyapunov function. π΅ = ππ π−1 + π ππ[2] −1 π , ππ₯ (A.6) and the matrix ππ is obtained by replacing each entry πππ of π by its derivative in the direction of π, ππππ . The quantity π(π΅) is the Lozinskii Μ measure of π΅ with respect to a vector norm | ⋅ | in π π, π = ( π2 ), defined by π (π΅) = lim+ β→0 |πΌ + βπ΅| − 1 , β (A.7) see [14, p. 41]. It is shown in [8] that, if π· is simply connected, the condition π2 < 0 rules out the presence of any orbit that gives rise to a simple closed rectifiable curve that is invariant for (A.1), such as periodic orbits, homoclinic orbits, and heteroclinic cycles. Moreover, it is robust under πΆ1 local perturbations of π near any nonequilibrium point that is nonwandering. In particular, the following global stability result is proved in Theorem 3.5 of [8]. Theorem A.4. Assume that π· is simply connected and that assumptions (H1), (H2) hold. Then the unique equilibrium π₯ of (A.1) is globally stable in π· if π2 < 0. Acknowledgments The authors are very grateful to two referees for their valuable comments and suggestions which led to an improvement of their original paper. They also thank the Handling Editor and all the editorial staff for the help and support provided to them. This work is supported by the NSF of China (no. 11271314) and NSF of Henan Province (no. 112300410058). References [1] Y. Xiao and S. Tang, “Dynamics of infection with nonlinear incidence in a simple vaccination model,” Nonlinear Analysis. Real World Applications, vol. 11, no. 5, pp. 4154–4163, 2010. [2] L.-M. Cai and X.-Z. Li, “Global analysis of a vector-host epidemic model with nonlinear incidences,” Applied Mathematics and Computation, vol. 217, no. 7, pp. 3531–3541, 2010. [3] Z. Hu, W. Ma, and S. Ruan, “Analysis of SIR epidemic models with nonlinear incidence rate and treatment,” Mathematical Biosciences, vol. 238, no. 1, pp. 12–20, 2012. 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